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Almost all grass has a hard time growing in the shade, but zoysia has a harder time with it. It can survive, but rarely thrives. However, you can take a look at Zeon and El Toro zoysia. They are considered the most shade tolerant.

Zoysia is a bit more aggressive than Bermuda.

Zoysia is pretty hardy, so I wouldn't worry too much, especially since the spot is not very large. If you want, you could snip away the oil saturated grass to make way for new (clean) growth. You could also douse the area with soapy water (use dish soap) to help loosen the oil particles.

This article will help with that: https://www.gardeningknowhow.com/lawn-care/specific/zoysia/controlling-zoysia-grass.htm

The ideal soil pH for zoysia grass is typically is 6 to 7.

It is likely a fungus. Here is information on a few common ones:
https://www.gardeningknowhow.com/lawn-care/lgen/grass-fungus.htm